Sense and nonsense in photon structure Ji Chla

Sense and nonsense in photon structure Jiří Chýla Institute of Physics, Prague Ø Ø Ø Ø The nature of PDF of the photon Why semantics matters Counting the powers of αs Factorization scale and scheme (in)independence What is wrong with DIS factorization scheme? QCD analysis of photon structure function Numerical results (J. Hejbal) J. Chýla: JHEP 04 (2000)007 J. Chýla: hep-ph/05 J. Hejbal: talk at this Conference 10/7/2020 Jiří Chýla 1

common Sense and nonsense in photon structure sense Jiří Chýla Institute of Physics, Prague Ø Ø Ø Ø The nature of PDF of the photon Why semantics matters Counting the powers of αs Factorization scale and scheme (in)independence What is wrong with DIS factorization scheme? QCD analysis of photon structure function Numerical results (J. Hejbal) J. Chýla: JHEP 04 (2000)007 J. Chýla: hep-ph/05 J. Hejbal: talk at this Conference 10/7/2020 Jiří Chýla 2

Basic facts Colour coupling: is actually a function of renormalization scale μ as well as renormalization scheme RS: At NLO renormalization scheme can be labeled by ΛRS 10/7/2020 Jiří Chýla 3

Parton distribution functions: quark singlet and nonsinglet densities as well as gluon density depend on factorization scale M and factorization scheme FS, i. e. renormalization scale μ and factorization scale M are independent parameters that should not be identified! 10/7/2020 Jiří Chýla 4

Evolution equations: inhomogeneous terms specific to photon 10/7/2020 Jiří Chýla 5

Splitting functions: homogeneous: inhomogeneous: where comes from pure QED! free, defining the FS unique, 10/7/2020 Jiří Chýla 6

General solution of evolution equations: Point-like part (PL): particular solution of the full inhomogeneous equation resulting from resummation of the series of diagrams starting with pure QED: 10/7/2020 Hadronic part (HAD): general solution of the corresponding homogeneous one Jiří Chýla 7

All the difference between hadron and photon structure functions comes from the pointlike part of PDF of the photon. Only the pointlike solutions of the evolution equations will be therefore considered. standard, but wrong interpretation: as switching off QCD by sending we get i. e. as expected the pure QED contribution! 10/7/2020 Jiří Chýla 8

easy way to wrong interpretation of the correct result: start from pure QED (in units of α/2π) replace integrating derivatives using define boundary condition voilà: but it is clear that this is just mirage as q is of pure QED nature and has nothing to do with QCD! 10/7/2020 Jiří Chýla 9

Asymptotic pointlike solution for asymptotic values of M 10/7/2020 Jiří Chýla 10

Structure function: where the coefficient functions describe 10/7/2020 pure QED Jiří Chýla QCD effects 11

Why semantics matters To avoid confusion, we should agree on the meaning of the terms leading and next-to-leading order. Recall their meaning for QED part where contains QCD effects For this quantity the QED contribution is subtarcted and the terms leading and next-to-leading orders are applied to QCD contribution r(Q) only. This procedure that should be adopted for other physical quantities as well, including 10/7/2020 Jiří Chýla 12

Counting powers of αs What does it mean that some quantity behaves as ? The standard claim that PDF of the photon behave as is inconsistent with the fact that switching off QCD we get, as we must, pure QED results for the pointlike part of photon structure function. 10/7/2020 Jiří Chýla 13

Factorization scale and scheme independence recall the situation for proton structure function the scale and scheme independence implies which in turm means that at NLO we get where FS invariant or 10/7/2020 is related to the nonuniversal NLO splitting function P(1). can be chosen freely to label different factorization schemes. Jiří Chýla 14

So either P(1) or C(1), but not both independently, can be chosen at will to specify the FS. For instance, F(Q) can be written as a function of C(1) explicitly as Mechanism guaranteeing FS invariance of F(Q): Choice of C(1) here is compensated by change of C(1) here MSbar: used for technical reasons DIS: Fp=q to all orders ZERO: EE in LO form to all orders 10/7/2020 Jiří Chýla 15

Factorization scale and scheme invariance of photon structure function For the pointlike part of the nonsinglet quark distribution function of the photon the situation is more complicated as the expression involves photonic coefficient function Factorization scale invariance the leads to the conclusion that the first non-universal inhomogeneous splitting function k(1) is, similarly as P(1), func-tion of C(1) (or the other way around). So C(1) can again be used to label the factorization scale ambiguity. 10/7/2020 Jiří Chýla 16

The fact that photon structure function behaves as. follows directly from factorization scale independence: As this expression is independent of M, we can take any M to evaluate it, for instance M 0. For M=M 0 the first term in vanishes and we get for the r. h. s. i. e. manifestly the expansion which starts with O(α) pure QED contribution and includes standard QCD corrections. This expansions vanishes when QCD is switched off and there is no trace of the supposed behaviour. 10/7/2020 Jiří Chýla 17

What is wrong with DIS FS? Recall Moch, Vermaseren, Vogt in Photon 05: 10/7/2020 Jiří Chýla 18

In the standard approach the lowest order, purely QED photonic coefficient function C(0), is treated in the same way as genuine QCD NLO coefficient function C(1) , i. e. is “absorbed” in the definition of PDF of the photon in the so called “DISγ” factorization scheme, with the same boundary condition But there is no mechanism guaranteeing the invariance of photon structure function under this change. Note that such mechanism must hold also for pure QED contribution as it must operate at any fixed order and cannot thus mix orders of QED and QCD. 10/7/2020 Jiří Chýla 19

However, getting rid of the troubling term via this redefinition violates factorization scheme invariance. The QED box diagram regularized by quark mass mq gives defines the QED part of the quark distribution function of the photon However, redefining the quark distribution function via the substitution 10/7/2020 Jiří Chýla 20

does not change the evolution equation in f-scheme Fundamental difference with respect to redefinition procedure in QCD Keeping the sum f-independent implies But we cannot impose on the same boundary condition in all f-schemes!! since we would get manifestly f-dependent expression 10/7/2020 Jiří Chýla 21

QCD analysis of photon structure function dropping charge factors we can separate contributions of individual orders of QCD in the following manner 10/7/2020 Jiří Chýla 22

Comparison of the terms included in the standard definition of the LO and NLO approximations with those of mine: Note, in particular, that the in standard approach the pure QED coefficient function enters first at the NLO approximation 10/7/2020 Jiří Chýla 23

Conclusions The organization of finite order QCD approximations of the photon structure function which follows closely that of the e+e- annihilation to hadrons and separates the pure QED contribution has been discussed. It differs significantly from the standard one by the set of terms included at each finite order. As shown in the talk of J. Hejbal, this difference is sizable and of clear phenomenological relevance. 10/7/2020 Jiří Chýla 24